Math 54, midterm 1
Instructor: Semyon Dyatlov
July 9, 2010
Name:
SID:
Problem 1:
/ 10
Problem 2:
/ 10
Problem 3:
/ 10
Problem 4:
/ 10
Problem 5:
/ 10
Total:
/ 50
• Write your solutions in the space provided. Do not use your own paper. I can give
you extra paper if needed. Indicate clearly where your answer is.
• Explain your solutions as clearly as possible. This will help me find what you did
right and what you did wrong, and award partial credit if possible.
• Justify all your steps. (Problem 3 is exempt from this rule.) A correct answer with
no justification will be given 0 points. Pictures without explanations are not counted
as justification. You may cite a theorem from the book by stating what it says.
• No calculators or notes are allowed on the exam, except for a single two-sided 5”×9”
sheet of hand-written notes. Cheating will result in academic and/or disciplinary
action. Please turn off cellphones and other electronic devices.
1. (a) Find the standard matrix of the transformation T : R3 → R2 given by the formula
T (x1 , x2 , x3 ) = (x1 + 2x2 , x2 + 2x3 ).
1
(b) Describe the solution set of the equation T (~x) =
in parametric vector form.
3
2. (a) Let A be a 3 × 3 matrix such that A−1


1 2 3
= 4 5 6  . Solve the equation
7 8 10
 
1
A~x = 0 .
1
(b) Find a linear dependence relation between the vectors
2
1
3
~a1 =
, ~a2 =
, ~a3 =
.
4
1
6
3. Circle the correct answer for each of the following true/false or multiple choice
questions, or write in the correct answer in the blank space provided. No justification is
required. For each question, you get 1 point if you answer it correctly, 0 points if you
choose not to answer, and −1 point if you answer it incorrectly. (If your total score for the
problem is negative, it will be changed to zero.)
(a) For each matrix A, det(AT A) ≥ 0.
True
False
(b) For all square matrices A and B of the same size, if AB = 0, then A = 0 or B = 0.
True
False
(c) For all square matrices A and B of the same size, det(A2 − B 2 ) = det(A + B) ·
det(A − B).
True
False
(d) If the equation A~x = ~b has a solution for each ~b, then A is square and det A 6= 0.
True
False
(e) If AB is well-defined and invertible and A is a square matrix, then B is invertible.
True
False
(f) If A has size m × n and B has size
× k.
×
, then AB is well-defined and has size
(g) If a linear transformation T : Ra → Rb is onto, then:
a≤b
a≥b
(h) If the vectors ~a1 , ~a2 , ~a3 ∈ R4 form a linearly dependent set, then each of these vectors
is a linear combination of the other two.
True
False
(i) For each matrix A, if the equation A~x = ~0 has unique solution, then for each ~b, the
number of solutions to the equation A~x = ~b is either 0 or 1.
True
False
(j) If A is a square matrix and AT A = I, then AAT = I.
True
False
1 0
1 4
4. (a) Find all t ∈ R such that the matrix A =
+t
has linearly dependent
1 1
0 1
columns.

1
0
(b) Find det 
0
1
2
0
4
3
0
2
0
0

3
0
.
5
5
5. Solve one of the following two problems. Mark which one you want graded.
(a) Assume that A and B are square matrices of the same size and A2 = B 2 = (AB)2 =
I. Show that AB = BA.
(b) Let ~a1 , ~a2 , ~a3 ∈ Rn . Prove that Span{~a1 , ~a2 } ⊂ Span{~a1 , ~a2 , ~a3 }. Here the sign
‘⊂’ means ‘is contained in’. Also, prove that if ~a3 ∈ Span{~a1 , ~a2 }, then Span{~a1 , ~a2 } =
Span{~a1 , ~a2 , ~a3 }.